Question

Water is poured into a container that has a leak. The mass m of the water is given as a function of time t by m = 4.8t0.8 - 2.7t + 16, with t ≥ 0, m in grams, and t in seconds. (a) At what time is the water mass greatest, and (b) what is that greatest mass? What is the rate of mass change at (c) t = 2.3 s and (d) t = 5.2 s?

Answer 1

To find the time when the water mass is greatest, one must calculate the derivative of the mass function with respect to time, set it equal to zero, and solve for time. The greatest mass is then determined by substituting this time back into the mass function. For the rate of mass change at specific times, the derived rate of change function is used.

Step-by-step explanation:

To determine at what time the mass of water, m, is greatest in the container with a leak, we need to find the maximum point of the given function m = 4.8t^0.8 - 2.7t + 16. To achieve this, we take the derivative of m with respect to t and set it equal to zero to find critical points that could be our maximum. After calculating the derivative, we need to solve for t when dm/dt = 0.

For part (b), we evaluate the original function at the time t found in part (a) to find the greatest mass.

To find the rate of mass change at specific times (c) t = 2.3 s and (d) t = 5.2 s, we simply plug these values into the derivative of the mass function that we calculated earlier.